Problem 68 - Intermediate Algebra

Question

The orbits of stars, planets, comets, asteroids, and satellites all have the shape of one of the conic sections. Astronomers use a measure called eccentricity to describe the shape and elongation of an orbital path. For the circle and ellipse, eccentricity e is calculated with the formula \(e=\frac{c}{d},\) where \(c^{2}=\left|a^{2}-b^{2}\right|\) and \(d\) is the larger value of a or b. For a hyperbola, eccentricity e is calculated with the formula \(e=\frac{c}{d},\) where \(c^{2}=a^{2}+b^{2}\) and the value of \(d\) is equal to a if the hyperbola has \(x\) -intercepts or equal to b if the hyperbola has \(y\) -intercepts. A. \(\frac{x^{2}}{36}-\frac{y^{2}}{13}=1\) B. \(\frac{x^{2}}{4}+\frac{y^{2}}{4}=1\) C. \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) D. \(\frac{y^{2}}{25}-\frac{x^{2}}{39}=1\) G. \(\frac{x^{2}}{16}-\frac{y^{2}}{65}=1\) E. \(\frac{x^{2}}{17}+\frac{y^{2}}{81}=1\) F. \(\frac{x^{2}}{36}+\frac{y^{2}}{36}=1\) H. \(\frac{x^{2}}{144}+\frac{y^{2}}{140}=1\) What do you notice about the values of \(e\) for the equations you identified as hyperbolas?

Step-by-Step Solution

Answer

The eccentricity \(e\) for the hyperbolas is always greater than 1.

Step 1: Identify conic sections

Start by identifying the type of conic section of each equation. An equation of the form \( \frac{x^{2}}{a^2} - \frac{y^{2}}{b^2} = 1 \) or \( \frac{y^{2}}{a^2} - \frac{x^{2}}{b^2} = 1 \) represents a hyperbola. Equations of the form \( \frac{x^{2}}{a^2} + \frac{y^{2}}{b^2} = 1 \) represent ellipses, and if \(a = b\), they represent circles.

Step 2: Determine eccentricity for hyperbolas

For hyperbolas, use the formula \( e = \frac{c}{d} \), where \( c^2 = a^2 + b^2 \) and \( d \) is determined by the orientation of the hyperbola (\(a\) if \(x\)-intercepts, \(b\) if \(y\)-intercepts). Calculate \(e\) for each hyperbola.

Step 3: Calculate eccentricity for ellipses

For ellipses, the formula is \( e = \frac{c}{d} \), where \( c^2 = \left|a^{2} - b^{2}\right| \) and \( d \) is the larger of \(a\) or \(b\). Calculate \(e\) for each ellipse and circle.

Step 4: Identify equations

From the equations provided: A and D are hyperbolas since they match the form for hyperbolas. B, C, E, F, and H are ellipses, where F is a circle since \(a = b\). G is a hyperbola as well. Identify each correctly before calculations.

Key Concepts

Eccentricity

Eccentricity is a crucial concept in understanding the geometry of conic sections, such as circles, ellipses, and hyperbolas, especially in astronomy. This value, denoted as \( e \), measures how much these conics deviate from being circular. In simple terms:

• An eccentricity of \( e = 0 \) indicates a perfect circle.
• For ellipses, \( 0 < e < 1 \), showing varying degrees of elongation.
• Hyperbolas have \( e > 1 \), indicating they are more open.

In both ellipses and hyperbolas, eccentricity is calculated using different formulas based on their specific geometric properties, emphasizing their distinct shapes. Understanding eccentricity helps astronomers and mathematicians predict how objects like planets will travel in their orbits around other celestial bodies.

Ellipses

Ellipses are one of the central types of conic sections with two focal points. They have an oval shape and are described mathematically by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here's what you should know about ellipses:

• An ellipse is defined by two main parameters: the semi-major axis \(a\) and the semi-minor axis \(b\).
• If \(a = b\), the ellipse becomes a circle.
• The further apart the foci, the more elongated the ellipse.

In terms of eccentricity, the ellipse’s elongation increases with \( e \), but it always stays between 0 and 1. This property makes ellipses excellent models for planetary orbits and any motions that require looping paths around a central mass.

Hyperbolas

Hyperbolas are conic sections that emerge when a plane intersects both nappes of a double cone. These curves have two arms that extend infinitely and are represented by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

• They are defined by the distances between each vertex and their respective foci.
• Unlike ellipses, hyperbolas open along two branches, resembling mirror images.
• The eccentricity \( e \) of a hyperbola is always greater than 1, which helps to quantify its openness.

In context with astronomy, hyperbolas can describe the paths of comets and other celestial bodies that only pass by a star once. Due to their high eccentricity, these paths do not loop, illustrating their path without return, unless gravitational interactions alter their trajectory.

Circles

Circles are the simplest form of conic sections, characterized by complete symmetry about their center. When an ellipse has equal semi-major and semi-minor axes, it becomes a circle.

• The equation of a circle is \( \frac{x^2}{r^2} + \frac{y^2}{r^2} = 1 \), where \( r \) is the radius.
• The eccentricity \( e \) of a circle is 0, reflecting the fact it has perfect symmetry and no elongation.

In celestial mechanics, when an object's orbit is perfectly circular, it indicates equal gravitational forces in all directions from the central body. This is quite rare in natural celestial orbits but models simple predictable circular motion for satellites and other human-made objects.

Orbits

Orbits are the paths that celestial bodies like planets, moons, and man-made satellites follow around larger bodies such as stars and planets. The type of conic section an orbit takes depends on the energy and velocity of the orbiting body.

• Elliptical orbits, with \( 0 < e < 1 \), are the most common in our solar system, describing paths around the sun and planets.
• Circles \( e = 0 \) are a special case of elliptical orbits with no eccentricity.
• Hyperbolic orbits \( e > 1 \) indicate a celestial body is on a trajectory to leave the gravitational pull of the central body.

Understanding orbits is critical for space exploration, navigation, and predicting celestial events. These paths highlight how gravity shapes the motion of objects across the universe, influenced by both their eccentricity and conic section type.